differential geometry

synthetic differential geometry

Axiomatics

• smooth topos

  • infinitesimal space

    • amazing right adjoint

  • Kock-Lawvere axiom

  • integration axiom

  • microlinear space

• synthetic differential supergeometry

  • super smooth topos

• . infinitesimal cohesion

  • de Rham space

  • formally smooth morphism, formally unramified morphism, formally étale morphism

  • jet bundle

Models

• Models for Smooth Infinitesimal Analysis

  • Fermat theory

  • smooth algebra ($C^{\mathrm{\infty }}$-ring)

  • smooth locus

  • smooth manifold, formal smooth manifold, derived smooth manifold

  • smooth space, diffeological space, Frölicher space

  • smooth natural numbers

• Cahiers topos

• smooth ∞-groupoid

• synthetic differential ∞-groupoid

Concepts

• tangent bundle,

  • vector field, tangent Lie algebroid;

  • differentiation, chain rule

• differential forms

• differential equation, variational calculus

  • Euler-Lagrange equation, de Donder-Weyl formalism, variational bicomplex, phase space

• connection on a bundle, connection on an ∞-bundle

• Riemannian manifold

  • isometry, Killing vector field

Theorems

• Hadamard lemma

• Borel's theorem

• Boman's theorem

• Steenrod-Wockel approximation theorem

• Poincare lemma

• Stokes theorem

• de Rham theorem

• Chern-Weil theory

Applications

• Lie theory, ∞-Lie theory

• Chern-Weil theory, ∞-Chern-Weil theory

  • gauge theory

  • ∞-Chern-Simons theory

• Klein geometry, Klein 2-geometry, higher Klein geometry

• Euclidean geometry, Cartan geometry, higher Cartan geometry

• Riemannian geometry

  • gravity, supergravity

physics, mathematical physics

Surveys, textbooks and lecture notes

• (higher) category theory and physics

• geometry of physics

• books and reviews, physics resources

theory (physics), model (physics)

• mechanics

  • mass, charge, momentum, angular momentum, moment of inertia

  • dynamics on Lie groups

    • rigid body dynamics

• field (physics)

  • Lagrangian mechanics

    • configuration space, state

    • action functional, Lagrangian

    • covariant phase space, Euler-Lagrange equations

  • Hamiltonian mechanics

    • phase space

    • symplectic geometry

      • Poisson manifold

      • symplectic manifold

      • symplectic groupoid

    • multisymplectic geometry

      • n-symplectic manifold

  • spacetime

    • smooth Lorentzian manifold

    • special relativity

    • general relativity

    • gravity

      • supergravity, dilaton gravity

      • black hole

• Classical field theory

  • classical physics

    • classical mechanics
    • waves and optics
    • thermodynamics

• Quantum Mechanics

  • in terms of †-compact categories

  • quantum information

  • Hamiltonian operator

  • density matrix

  • Kochen-Specker theorem

  • Bell's theorem

  • Gleason's theorem

• Quantization

  • geometric quantization

  • deformation quantization

  • path integral quantization

  • semiclassical approximation

• Quantum Field Theory

  • Axiomatizations

    • algebraic QFT

      • Wightman axioms

      • Haag-Kastler axioms

        • operator algebra

        • local net

          • conformal net

      • Reeh-Schlieder theorem

      • Osterwalder-Schrader theorem

      • PCT theorem

      • Bisognano-Wichmann theorem

        • modular theory

      • spin-statistics theorem

        • boson, fermion

    • functorial QFT

      • cobordism

      • (∞,n)-category of cobordisms

      • cobordism hypothesis-theorem

      • extended topological quantum field theory

  • Tools

    • perturbative quantum field theory, vacuum

    • effective quantum field theory

    • renormalization

    • BV-BRST formalism

    • geometric ∞-function theory

  • particle physics

    • phenomenology

    • models

      • standard model of particle physics

      • fields and quanta

      • Grand Unified Theories, MSSM

    • scattering amplitude

      • on-shell recursion, KLT relations

  • Structural phenomena

    • universality class

    • quantum anomaly

      • Green-Schwarz mechanism

    • instanton

    • spontaneously broken symmetry

    • Kaluza-Klein mechanism

    • integrable systems

    • holonomic quantum fields

  • Types of quantum field thories

    • TQFT

      • 2d TQFT

      • Dijkgraaf-Witten theory

      • Chern-Simons theory

    • TCFT

      • A-model, B-model

      • homological mirror symmetry

    • QFT with defects

    • conformal field theory

    • (1,1)-dimensional Euclidean field theories and K-theory

    • (2,1)-dimensional Euclidean field theory and elliptic cohomology

    • CFT

      • WZW model

      • 6d (2,0)-supersymmetric QFT

    • gauge theory

      • field strength

      • gauge group, gauge transformation, gauge fixing

      • examples

        • electromagnetic field, QED

          • electric charge
          • magnetic charge

        • Yang-Mills field, QCD

          • Yang-Mills theory
          • spinors in Yang-Mills theory
          • topological Yang-Mills theory

        • Kalb-Ramond field

        • supergravity C-field

        • RR field

        • first-order formulation of gravity

          • general covariance
          • supergravity * D'Auria-Fre formulation of supergravity
          • gravity as a BF-theory

    • sigma-model

      • particle, relativistic particle, fundamental particle, spinning particle, superparticle

      • string, spinning string, superstring

      • membrane

      • AKSZ theory

• String Theory

  • string theory results applied elsewhere

equivalence

• equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)

• identity type, equivalence in homotopy type theory

• isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category

• natural equivalence, natural isomorphism

• gauge equivalence

• Examples.

  • equivalence of categories, adjoint equivalence, weak equivalence of internal categories, Morita equivalence, equivalence of 2-categories, equivalence of (∞,1)-categories

principle of equivalence

• univalence

equation

• fiber product, pullback

• homotopy pullback

• Examples.

  • linear equation, differential equation, ordinary differential equation, critical locus

  • Euler-Lagrange equation, Einstein equation, wave equation

  • Schrödinger equation, Knizhnik-Zamolodchikov equation, Maurer-Cartan equation, quantum master equation, Euler-Arnold equation, Fuchsian equation, Fokker-Planck equation, Lax equation